Introduction to Statistics for Astronomers and Physicists

Section 0: Introduction, Lecture Outline, Course Format

Angus H Wright

2021-03-22

Setup & Course Outline

This course will be taught in 4 parts, each spanning from 2-4 weeks

Section 1: Data Description, Analysis, and Modelling (Weeks 1-2)

Topics include:

##Section 2: Probability & Decision Making (Weeks 3-5)

Topics include:

##Section 3: Bayesian Statistics (Weeks 6-8

Topics include:

##Section 4: Parameter Simulation, Optimisation, and Analysis (Weeks 9-12)

Topics include:

Course Philosophy

This course is designed to be a practical introduction to statistics for astronomers and physicists who are starting their research careers and have had little (or perhaps no) previous education in statistics and statistical data analysis. The course (and these lecture notes) are not designed to be a statistics reference text. Rather the material presented here is designed to guide students on a suitable path towards robust data analysis and research.

The course will present many aspects of data analysis that are widely relevant to modern astronomy and physics. We will borrow heavily from standard statistical problems and thought experiments in an effort to convey important points, and elucidate common statistical and logical fallacies. Problems will almost always be explored using a mixture of tools simultaneously: plain English, math, computer code, graphs, and more.

Rmarkdown

Slides and lecture notes for this course are prepared in Rmarkdown, and provided to you after the lectures.

The utility of Rmarkdown is that it allows running execution of code chunks alongside markdown-style text, in a wide array of languages, including R, Python, bash, Rcpp, javascript, and more. This allows us to present examples in multiple languages easily within one document. For example, if I want to plot a function, I can do so in:
#in R
theta=seq(0,2,by=0.01)
sinu=1+sin(2*pi*theta)
magplot(theta,sinu,type='l')
#or in python
import numpy as np
import matplotlib.pyplot as plt
theta=np.arange(0.,2.,0.01)
sinu=1+np.sin(2*np.pi*theta)
plt.plot(theta,sinu)
plt.show()
Information generated and stored within blocks is persistent, and code-blocks with different engines can also cross-communicate. This means that, for example, we can:
#Create some data in R
#E.g. random draws from f~N(0,1)
x=rnorm(1e3) 
y=rnorm(1e3) 
#Default plot in R
plot(x,y)
#and access it directly in python
plt.scatter(r.x,r.y)
plt.show()

Modern Science and the Requirement of Programming

Modern physics and astronomy requires an understanding of programming. From theoreticians writing models to experimentalists writing analysis pipelines, most physicists and astronomers will use read, write, or use a computer program every day.

The ubiquitousness of programming in the modern physical sciences is intimately linked to the important role that statistics plays in these fields. Modern science is increasingly reliant on large and/or complex datasets, which fundamentally must be analysed using modern statistical methods. A classic example is model optimisation (which we will cover in detail in this course): let us take a relatively simple dataset that we know follows a generic beta distribution, and attempt to model this dataset using a function containing 2 parameters:

\[ Y = Beta(X, \alpha,\beta) + {\rm noise} \\ 0\leq X \leq 1 \\ \alpha, \beta \in [0,\infty) \]

One might be inclined to attempt to fit a model to these data by-hand, using trial and error:

Using this approach we can get a reasonable fit with \(\alpha=2.8,\beta=6.7\):

But is this solution near the truth? How close is good enough? And what are the uncertainties on the parameters? These are all important in modern science, and they are precisely the sort of questions/problems that computers/programs are designed to tackle. With just one command, we can use R/python to reach a more accurate solution, in a fraction of the time.

#Estimate parameters using Nonlinear Least Squares (nls) in R 
fit_R=nls(y~dbeta(x,alpha,beta), #The function to fit
          data=list(x=x,y=y),    #The data 
          start=list(alpha=2,beta=5), #The initial guess
          algorithm='port',      #The algorithm 
          lower=c(0,0))          #The lower bounds
best_R=summary(fit_R)$parameters
#Estimate parameters using scipy.optimize.curve_fit in python
import scipy.optimize
import scipy.stats
best_py, cov_py = scipy.optimize.curve_fit(
         scipy.stats.beta.pdf, #The function to fit
         r.x, r.y,         #The data
         p0=[2,5],         #The initial guess 
         bounds=(0,np.inf),#The lower and upper bounds
         method='trf')     #The fitting algorithm

Obviously these fits are superior to those which we can reach by-hand in terms of accuracy, effort, and runtime. But the most important benefit is in terms of uncertainty estimation. Statistical computing is important for a wide range of reasons, but arguably the first and most important reason is for the computation of measures of uncertainty.

#Model statistics in R
summary(fit_R)
## 
## Formula: y ~ dbeta(x, alpha, beta)
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## alpha  3.21166    0.01453   221.0   <2e-16 ***
## beta   7.70586    0.03728   206.7   <2e-16 ***
## ---
## Signif. codes:  
## 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.09854 on 998 degrees of freedom
## 
## Algorithm "port", convergence message: both X-convergence and relative convergence (5)
#Model parameters and covariance in python
print(best_py,cov_py) 
## [3.21166224 7.70585894] [[0.00021118 0.00050005]
##  [0.00050005 0.00138966]]

Do I need to know R or Python?

Despite what the internet will tell you, today there is very little separating the two languages in terms of functionality. Both languages can be run as a subprocess of the other, both have well developed tools for data analysis and machine learning, and both have a wealth of tutorials and guides to help new users enter the game. Overall, there is one major consideration that will (and should) drive your choice of which language to learn first: what languages do your colleagues use? Any perceived benefit or detriment of the languages will invariably be overwhelmed by whether or not you are able to share and discuss code together with your colleagues. So, this is mostly a case where joining the herd is probably the sensible choice.

##What’s the difference?

R was originally developed as a statistics and data analysis language, and so many data analysis tools are available natively within base R. Additional tools are available through the Comprehensive R Archive Network (CRAN), which contains over 10k packages that are all required to be fully documented. This means that if you want to perform a particular flavour of statistical analysis, there is a good chance that well developed code already exists within R to do it (or at least to get you started).

Python’s strength lies within the its use as a general programming language beyond data analysis. In comparison, python is Personally I code primarily in R, however a significant R and Python are

In practice, the vast majority of examples in this course will be programmed in R. However, as you can see from the above, if you can understand one, then you can probably understand both. In this lecture we will go through some examples of R and python code, so that you have an introduction to the important parts (and can follow along without much trouble).

Lecture 0: Introduction to Statistics

for (i in 1:10) plot(runif(1000), ylim = c(0, 1)) # for example

x = 5  # radius of a circle

For a circle with the radius 5, its area is 78.5398163.

Why do we need statistics?

print(iris) 
##     Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1            5.1         3.5          1.4         0.2
## 2            4.9         3.0          1.4         0.2
## 3            4.7         3.2          1.3         0.2
## 4            4.6         3.1          1.5         0.2
## 5            5.0         3.6          1.4         0.2
## 6            5.4         3.9          1.7         0.4
## 7            4.6         3.4          1.4         0.3
## 8            5.0         3.4          1.5         0.2
## 9            4.4         2.9          1.4         0.2
## 10           4.9         3.1          1.5         0.1
## 11           5.4         3.7          1.5         0.2
## 12           4.8         3.4          1.6         0.2
## 13           4.8         3.0          1.4         0.1
## 14           4.3         3.0          1.1         0.1
## 15           5.8         4.0          1.2         0.2
## 16           5.7         4.4          1.5         0.4
## 17           5.4         3.9          1.3         0.4
## 18           5.1         3.5          1.4         0.3
## 19           5.7         3.8          1.7         0.3
## 20           5.1         3.8          1.5         0.3
## 21           5.4         3.4          1.7         0.2
## 22           5.1         3.7          1.5         0.4
## 23           4.6         3.6          1.0         0.2
## 24           5.1         3.3          1.7         0.5
## 25           4.8         3.4          1.9         0.2
## 26           5.0         3.0          1.6         0.2
## 27           5.0         3.4          1.6         0.4
## 28           5.2         3.5          1.5         0.2
## 29           5.2         3.4          1.4         0.2
## 30           4.7         3.2          1.6         0.2
## 31           4.8         3.1          1.6         0.2
## 32           5.4         3.4          1.5         0.4
## 33           5.2         4.1          1.5         0.1
## 34           5.5         4.2          1.4         0.2
## 35           4.9         3.1          1.5         0.2
## 36           5.0         3.2          1.2         0.2
## 37           5.5         3.5          1.3         0.2
## 38           4.9         3.6          1.4         0.1
## 39           4.4         3.0          1.3         0.2
## 40           5.1         3.4          1.5         0.2
## 41           5.0         3.5          1.3         0.3
## 42           4.5         2.3          1.3         0.3
## 43           4.4         3.2          1.3         0.2
## 44           5.0         3.5          1.6         0.6
## 45           5.1         3.8          1.9         0.4
## 46           4.8         3.0          1.4         0.3
## 47           5.1         3.8          1.6         0.2
## 48           4.6         3.2          1.4         0.2
## 49           5.3         3.7          1.5         0.2
## 50           5.0         3.3          1.4         0.2
## 51           7.0         3.2          4.7         1.4
## 52           6.4         3.2          4.5         1.5
## 53           6.9         3.1          4.9         1.5
## 54           5.5         2.3          4.0         1.3
## 55           6.5         2.8          4.6         1.5
## 56           5.7         2.8          4.5         1.3
## 57           6.3         3.3          4.7         1.6
## 58           4.9         2.4          3.3         1.0
## 59           6.6         2.9          4.6         1.3
## 60           5.2         2.7          3.9         1.4
## 61           5.0         2.0          3.5         1.0
## 62           5.9         3.0          4.2         1.5
## 63           6.0         2.2          4.0         1.0
## 64           6.1         2.9          4.7         1.4
## 65           5.6         2.9          3.6         1.3
## 66           6.7         3.1          4.4         1.4
## 67           5.6         3.0          4.5         1.5
## 68           5.8         2.7          4.1         1.0
## 69           6.2         2.2          4.5         1.5
## 70           5.6         2.5          3.9         1.1
## 71           5.9         3.2          4.8         1.8
## 72           6.1         2.8          4.0         1.3
## 73           6.3         2.5          4.9         1.5
## 74           6.1         2.8          4.7         1.2
## 75           6.4         2.9          4.3         1.3
## 76           6.6         3.0          4.4         1.4
## 77           6.8         2.8          4.8         1.4
## 78           6.7         3.0          5.0         1.7
## 79           6.0         2.9          4.5         1.5
## 80           5.7         2.6          3.5         1.0
## 81           5.5         2.4          3.8         1.1
## 82           5.5         2.4          3.7         1.0
## 83           5.8         2.7          3.9         1.2
## 84           6.0         2.7          5.1         1.6
## 85           5.4         3.0          4.5         1.5
## 86           6.0         3.4          4.5         1.6
## 87           6.7         3.1          4.7         1.5
## 88           6.3         2.3          4.4         1.3
## 89           5.6         3.0          4.1         1.3
## 90           5.5         2.5          4.0         1.3
## 91           5.5         2.6          4.4         1.2
## 92           6.1         3.0          4.6         1.4
## 93           5.8         2.6          4.0         1.2
## 94           5.0         2.3          3.3         1.0
## 95           5.6         2.7          4.2         1.3
## 96           5.7         3.0          4.2         1.2
## 97           5.7         2.9          4.2         1.3
## 98           6.2         2.9          4.3         1.3
## 99           5.1         2.5          3.0         1.1
## 100          5.7         2.8          4.1         1.3
## 101          6.3         3.3          6.0         2.5
## 102          5.8         2.7          5.1         1.9
## 103          7.1         3.0          5.9         2.1
## 104          6.3         2.9          5.6         1.8
## 105          6.5         3.0          5.8         2.2
## 106          7.6         3.0          6.6         2.1
## 107          4.9         2.5          4.5         1.7
## 108          7.3         2.9          6.3         1.8
## 109          6.7         2.5          5.8         1.8
## 110          7.2         3.6          6.1         2.5
## 111          6.5         3.2          5.1         2.0
## 112          6.4         2.7          5.3         1.9
## 113          6.8         3.0          5.5         2.1
## 114          5.7         2.5          5.0         2.0
## 115          5.8         2.8          5.1         2.4
## 116          6.4         3.2          5.3         2.3
## 117          6.5         3.0          5.5         1.8
## 118          7.7         3.8          6.7         2.2
## 119          7.7         2.6          6.9         2.3
## 120          6.0         2.2          5.0         1.5
## 121          6.9         3.2          5.7         2.3
## 122          5.6         2.8          4.9         2.0
## 123          7.7         2.8          6.7         2.0
## 124          6.3         2.7          4.9         1.8
## 125          6.7         3.3          5.7         2.1
## 126          7.2         3.2          6.0         1.8
## 127          6.2         2.8          4.8         1.8
## 128          6.1         3.0          4.9         1.8
## 129          6.4         2.8          5.6         2.1
## 130          7.2         3.0          5.8         1.6
## 131          7.4         2.8          6.1         1.9
## 132          7.9         3.8          6.4         2.0
## 133          6.4         2.8          5.6         2.2
## 134          6.3         2.8          5.1         1.5
## 135          6.1         2.6          5.6         1.4
## 136          7.7         3.0          6.1         2.3
## 137          6.3         3.4          5.6         2.4
## 138          6.4         3.1          5.5         1.8
## 139          6.0         3.0          4.8         1.8
## 140          6.9         3.1          5.4         2.1
## 141          6.7         3.1          5.6         2.4
## 142          6.9         3.1          5.1         2.3
## 143          5.8         2.7          5.1         1.9
## 144          6.8         3.2          5.9         2.3
## 145          6.7         3.3          5.7         2.5
## 146          6.7         3.0          5.2         2.3
## 147          6.3         2.5          5.0         1.9
## 148          6.5         3.0          5.2         2.0
## 149          6.2         3.4          5.4         2.3
## 150          5.9         3.0          5.1         1.8
##        Species
## 1       setosa
## 2       setosa
## 3       setosa
## 4       setosa
## 5       setosa
## 6       setosa
## 7       setosa
## 8       setosa
## 9       setosa
## 10      setosa
## 11      setosa
## 12      setosa
## 13      setosa
## 14      setosa
## 15      setosa
## 16      setosa
## 17      setosa
## 18      setosa
## 19      setosa
## 20      setosa
## 21      setosa
## 22      setosa
## 23      setosa
## 24      setosa
## 25      setosa
## 26      setosa
## 27      setosa
## 28      setosa
## 29      setosa
## 30      setosa
## 31      setosa
## 32      setosa
## 33      setosa
## 34      setosa
## 35      setosa
## 36      setosa
## 37      setosa
## 38      setosa
## 39      setosa
## 40      setosa
## 41      setosa
## 42      setosa
## 43      setosa
## 44      setosa
## 45      setosa
## 46      setosa
## 47      setosa
## 48      setosa
## 49      setosa
## 50      setosa
## 51  versicolor
## 52  versicolor
## 53  versicolor
## 54  versicolor
## 55  versicolor
## 56  versicolor
## 57  versicolor
## 58  versicolor
## 59  versicolor
## 60  versicolor
## 61  versicolor
## 62  versicolor
## 63  versicolor
## 64  versicolor
## 65  versicolor
## 66  versicolor
## 67  versicolor
## 68  versicolor
## 69  versicolor
## 70  versicolor
## 71  versicolor
## 72  versicolor
## 73  versicolor
## 74  versicolor
## 75  versicolor
## 76  versicolor
## 77  versicolor
## 78  versicolor
## 79  versicolor
## 80  versicolor
## 81  versicolor
## 82  versicolor
## 83  versicolor
## 84  versicolor
## 85  versicolor
## 86  versicolor
## 87  versicolor
## 88  versicolor
## 89  versicolor
## 90  versicolor
## 91  versicolor
## 92  versicolor
## 93  versicolor
## 94  versicolor
## 95  versicolor
## 96  versicolor
## 97  versicolor
## 98  versicolor
## 99  versicolor
## 100 versicolor
## 101  virginica
## 102  virginica
## 103  virginica
## 104  virginica
## 105  virginica
## 106  virginica
## 107  virginica
## 108  virginica
## 109  virginica
## 110  virginica
## 111  virginica
## 112  virginica
## 113  virginica
## 114  virginica
## 115  virginica
## 116  virginica
## 117  virginica
## 118  virginica
## 119  virginica
## 120  virginica
## 121  virginica
## 122  virginica
## 123  virginica
## 124  virginica
## 125  virginica
## 126  virginica
## 127  virginica
## 128  virginica
## 129  virginica
## 130  virginica
## 131  virginica
## 132  virginica
## 133  virginica
## 134  virginica
## 135  virginica
## 136  virginica
## 137  virginica
## 138  virginica
## 139  virginica
## 140  virginica
## 141  virginica
## 142  virginica
## 143  virginica
## 144  virginica
## 145  virginica
## 146  virginica
## 147  virginica
## 148  virginica
## 149  virginica
## 150  virginica
print(iris) 
##     Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1            5.1         3.5          1.4         0.2
## 2            4.9         3.0          1.4         0.2
## 3            4.7         3.2          1.3         0.2
## 4            4.6         3.1          1.5         0.2
## 5            5.0         3.6          1.4         0.2
## 6            5.4         3.9          1.7         0.4
## 7            4.6         3.4          1.4         0.3
## 8            5.0         3.4          1.5         0.2
## 9            4.4         2.9          1.4         0.2
## 10           4.9         3.1          1.5         0.1
## 11           5.4         3.7          1.5         0.2
## 12           4.8         3.4          1.6         0.2
## 13           4.8         3.0          1.4         0.1
## 14           4.3         3.0          1.1         0.1
## 15           5.8         4.0          1.2         0.2
## 16           5.7         4.4          1.5         0.4
## 17           5.4         3.9          1.3         0.4
## 18           5.1         3.5          1.4         0.3
## 19           5.7         3.8          1.7         0.3
## 20           5.1         3.8          1.5         0.3
## 21           5.4         3.4          1.7         0.2
## 22           5.1         3.7          1.5         0.4
## 23           4.6         3.6          1.0         0.2
## 24           5.1         3.3          1.7         0.5
## 25           4.8         3.4          1.9         0.2
## 26           5.0         3.0          1.6         0.2
## 27           5.0         3.4          1.6         0.4
## 28           5.2         3.5          1.5         0.2
## 29           5.2         3.4          1.4         0.2
## 30           4.7         3.2          1.6         0.2
## 31           4.8         3.1          1.6         0.2
## 32           5.4         3.4          1.5         0.4
## 33           5.2         4.1          1.5         0.1
## 34           5.5         4.2          1.4         0.2
## 35           4.9         3.1          1.5         0.2
## 36           5.0         3.2          1.2         0.2
## 37           5.5         3.5          1.3         0.2
## 38           4.9         3.6          1.4         0.1
## 39           4.4         3.0          1.3         0.2
## 40           5.1         3.4          1.5         0.2
## 41           5.0         3.5          1.3         0.3
## 42           4.5         2.3          1.3         0.3
## 43           4.4         3.2          1.3         0.2
## 44           5.0         3.5          1.6         0.6
## 45           5.1         3.8          1.9         0.4
## 46           4.8         3.0          1.4         0.3
## 47           5.1         3.8          1.6         0.2
## 48           4.6         3.2          1.4         0.2
## 49           5.3         3.7          1.5         0.2
## 50           5.0         3.3          1.4         0.2
## 51           7.0         3.2          4.7         1.4
## 52           6.4         3.2          4.5         1.5
## 53           6.9         3.1          4.9         1.5
## 54           5.5         2.3          4.0         1.3
## 55           6.5         2.8          4.6         1.5
## 56           5.7         2.8          4.5         1.3
## 57           6.3         3.3          4.7         1.6
## 58           4.9         2.4          3.3         1.0
## 59           6.6         2.9          4.6         1.3
## 60           5.2         2.7          3.9         1.4
## 61           5.0         2.0          3.5         1.0
## 62           5.9         3.0          4.2         1.5
## 63           6.0         2.2          4.0         1.0
## 64           6.1         2.9          4.7         1.4
## 65           5.6         2.9          3.6         1.3
## 66           6.7         3.1          4.4         1.4
## 67           5.6         3.0          4.5         1.5
## 68           5.8         2.7          4.1         1.0
## 69           6.2         2.2          4.5         1.5
## 70           5.6         2.5          3.9         1.1
## 71           5.9         3.2          4.8         1.8
## 72           6.1         2.8          4.0         1.3
## 73           6.3         2.5          4.9         1.5
## 74           6.1         2.8          4.7         1.2
## 75           6.4         2.9          4.3         1.3
## 76           6.6         3.0          4.4         1.4
## 77           6.8         2.8          4.8         1.4
## 78           6.7         3.0          5.0         1.7
## 79           6.0         2.9          4.5         1.5
## 80           5.7         2.6          3.5         1.0
## 81           5.5         2.4          3.8         1.1
## 82           5.5         2.4          3.7         1.0
## 83           5.8         2.7          3.9         1.2
## 84           6.0         2.7          5.1         1.6
## 85           5.4         3.0          4.5         1.5
## 86           6.0         3.4          4.5         1.6
## 87           6.7         3.1          4.7         1.5
## 88           6.3         2.3          4.4         1.3
## 89           5.6         3.0          4.1         1.3
## 90           5.5         2.5          4.0         1.3
## 91           5.5         2.6          4.4         1.2
## 92           6.1         3.0          4.6         1.4
## 93           5.8         2.6          4.0         1.2
## 94           5.0         2.3          3.3         1.0
## 95           5.6         2.7          4.2         1.3
## 96           5.7         3.0          4.2         1.2
## 97           5.7         2.9          4.2         1.3
## 98           6.2         2.9          4.3         1.3
## 99           5.1         2.5          3.0         1.1
## 100          5.7         2.8          4.1         1.3
## 101          6.3         3.3          6.0         2.5
## 102          5.8         2.7          5.1         1.9
## 103          7.1         3.0          5.9         2.1
## 104          6.3         2.9          5.6         1.8
## 105          6.5         3.0          5.8         2.2
## 106          7.6         3.0          6.6         2.1
## 107          4.9         2.5          4.5         1.7
## 108          7.3         2.9          6.3         1.8
## 109          6.7         2.5          5.8         1.8
## 110          7.2         3.6          6.1         2.5
## 111          6.5         3.2          5.1         2.0
## 112          6.4         2.7          5.3         1.9
## 113          6.8         3.0          5.5         2.1
## 114          5.7         2.5          5.0         2.0
## 115          5.8         2.8          5.1         2.4
## 116          6.4         3.2          5.3         2.3
## 117          6.5         3.0          5.5         1.8
## 118          7.7         3.8          6.7         2.2
## 119          7.7         2.6          6.9         2.3
## 120          6.0         2.2          5.0         1.5
## 121          6.9         3.2          5.7         2.3
## 122          5.6         2.8          4.9         2.0
## 123          7.7         2.8          6.7         2.0
## 124          6.3         2.7          4.9         1.8
## 125          6.7         3.3          5.7         2.1
## 126          7.2         3.2          6.0         1.8
## 127          6.2         2.8          4.8         1.8
## 128          6.1         3.0          4.9         1.8
## 129          6.4         2.8          5.6         2.1
## 130          7.2         3.0          5.8         1.6
## 131          7.4         2.8          6.1         1.9
## 132          7.9         3.8          6.4         2.0
## 133          6.4         2.8          5.6         2.2
## 134          6.3         2.8          5.1         1.5
## 135          6.1         2.6          5.6         1.4
## 136          7.7         3.0          6.1         2.3
## 137          6.3         3.4          5.6         2.4
## 138          6.4         3.1          5.5         1.8
## 139          6.0         3.0          4.8         1.8
## 140          6.9         3.1          5.4         2.1
## 141          6.7         3.1          5.6         2.4
## 142          6.9         3.1          5.1         2.3
## 143          5.8         2.7          5.1         1.9
## 144          6.8         3.2          5.9         2.3
## 145          6.7         3.3          5.7         2.5
## 146          6.7         3.0          5.2         2.3
## 147          6.3         2.5          5.0         1.9
## 148          6.5         3.0          5.2         2.0
## 149          6.2         3.4          5.4         2.3
## 150          5.9         3.0          5.1         1.8
##        Species
## 1       setosa
## 2       setosa
## 3       setosa
## 4       setosa
## 5       setosa
## 6       setosa
## 7       setosa
## 8       setosa
## 9       setosa
## 10      setosa
## 11      setosa
## 12      setosa
## 13      setosa
## 14      setosa
## 15      setosa
## 16      setosa
## 17      setosa
## 18      setosa
## 19      setosa
## 20      setosa
## 21      setosa
## 22      setosa
## 23      setosa
## 24      setosa
## 25      setosa
## 26      setosa
## 27      setosa
## 28      setosa
## 29      setosa
## 30      setosa
## 31      setosa
## 32      setosa
## 33      setosa
## 34      setosa
## 35      setosa
## 36      setosa
## 37      setosa
## 38      setosa
## 39      setosa
## 40      setosa
## 41      setosa
## 42      setosa
## 43      setosa
## 44      setosa
## 45      setosa
## 46      setosa
## 47      setosa
## 48      setosa
## 49      setosa
## 50      setosa
## 51  versicolor
## 52  versicolor
## 53  versicolor
## 54  versicolor
## 55  versicolor
## 56  versicolor
## 57  versicolor
## 58  versicolor
## 59  versicolor
## 60  versicolor
## 61  versicolor
## 62  versicolor
## 63  versicolor
## 64  versicolor
## 65  versicolor
## 66  versicolor
## 67  versicolor
## 68  versicolor
## 69  versicolor
## 70  versicolor
## 71  versicolor
## 72  versicolor
## 73  versicolor
## 74  versicolor
## 75  versicolor
## 76  versicolor
## 77  versicolor
## 78  versicolor
## 79  versicolor
## 80  versicolor
## 81  versicolor
## 82  versicolor
## 83  versicolor
## 84  versicolor
## 85  versicolor
## 86  versicolor
## 87  versicolor
## 88  versicolor
## 89  versicolor
## 90  versicolor
## 91  versicolor
## 92  versicolor
## 93  versicolor
## 94  versicolor
## 95  versicolor
## 96  versicolor
## 97  versicolor
## 98  versicolor
## 99  versicolor
## 100 versicolor
## 101  virginica
## 102  virginica
## 103  virginica
## 104  virginica
## 105  virginica
## 106  virginica
## 107  virginica
## 108  virginica
## 109  virginica
## 110  virginica
## 111  virginica
## 112  virginica
## 113  virginica
## 114  virginica
## 115  virginica
## 116  virginica
## 117  virginica
## 118  virginica
## 119  virginica
## 120  virginica
## 121  virginica
## 122  virginica
## 123  virginica
## 124  virginica
## 125  virginica
## 126  virginica
## 127  virginica
## 128  virginica
## 129  virginica
## 130  virginica
## 131  virginica
## 132  virginica
## 133  virginica
## 134  virginica
## 135  virginica
## 136  virginica
## 137  virginica
## 138  virginica
## 139  virginica
## 140  virginica
## 141  virginica
## 142  virginica
## 143  virginica
## 144  virginica
## 145  virginica
## 146  virginica
## 147  virginica
## 148  virginica
## 149  virginica
## 150  virginica

Slide with Bullets

cars
cars

Slide with R Output

summary(cars)
##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

Slide with Plot

plot(pressure)